@Siddhant_Mudholkar
For second part you could divide the spring into two springs. Each will have a spring constant of 2k.
After equilibrium is attained, vertical force of surface tension = spring force at equilibrium
Further compressing or elongating the spring by a small amount, the restoring force is provided only by the extra length of spring.ma = - 2k x\omega = \sqrt{\cfrac{2k}{m}}

@pratyaksh_tyagi
But bro, if both rings are displaced by x then total extra length =2 x is ok but in equation shouldn't it be k x only because for first part we said that if total extension is x_0, we take x_0/2
So here for 2x we should take 2x/2 =x
Can you explain when we take x/2 and when 2x.